# Krieger's finite generator theorem for actions of countable groups III

## TL;DR

This paper advances the understanding of Rokhlin entropy for group actions by proving a finite generator theorem, exploring its properties, and relating it to other entropy notions, thereby deepening the theoretical framework of dynamical systems.

## Contribution

It introduces a non-ergodic finite generator theorem and establishes key properties and formulas for Rokhlin entropy, linking it to sofic and Kolmogorov--Sinai entropies.

## Key findings

- Proved a non-ergodic finite generator theorem.
- Established sub-additivity and semi-continuity of Rokhlin entropy.
- Connected Rokhlin entropy with sofic and classical entropy notions.

## Abstract

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semi-continuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov--Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.